Stochastic Model Predictive Control Approaches applied to

Drinking Water Networks

Juan M. Grosso1, Pablo Velarde2, Carlos Ocampo-Martinez1, Jose M. Maestre2

and Vicenc Puig1

1Automatic Control Department, Universitat Politecnica de Catalunya (UPC),Institut de Robotica i Informatica Industrial (CSIC-UPC),

Llorens i Artigas, 4-6, 08028 Barcelona, Spain.2Departamento de Ingenierıa de Sistemas y Automatica, Universidad de Sevilla,

Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain.

Abstract

Control of drinking water networks is an arduous task given their size and the presence of uncertaintyin water demand. It is necessary to impose different constraints for ensuring a reliable water supply inthe most economic and safe ways. To cope with uncertainty in system disturbances due to the stochasticwater demand/consumption, and optimize operational costs, this paper proposes three stochastic modelpredictive control (MPC) approaches, namely: chance-constrained MPC, tree-based MPC, and multiple-scenarios MPC. A comparative assessment of these approaches is performed when they are applied toreal case studies, specifically, a sector and an aggregate version of the Barcelona drinking water networkin Spain.

1 Introduction

Drinking water networks (DWNs) transport water from sources to consumers ensuring the quality of service[1]. Nevertheless, limited water sources, conservation and sustainability policies, as well as the infrastructurecomplexity for meeting consumer demands with appropriate flow pressure and quality levels make watermanagement a challenging problem [2]. Water demand forecasting based on historical data is commonlyused for the operational control of water supply along a given prediction horizon. However, the optimalityof such scheduling is affected by the one associated to water demand forecasts. Therefore, the scheduling ofcontrol inputs must be continuously adjusted. This leads to consider the DWNs as dynamical systems andtheir operation as optimal control problems, with the objective of satisfying water demands in an optimalmanner despite the presence of disturbances and uncertainties, and considering additionally issues such asconstraints on the manipulated and output variables and multiple conflicting control goals. Given the featuresof the problem, MPC provides a control framework capable of dealing with these issues in an explicit manner[2, 3]. The main idea of MPC is to obtain a control signal by solving, at each time step, a finite-horizonoptimization problem (FHOP) that takes into account a model of the system to predict its evolution and tosteer it in accordance to given objectives. The first component of the obtained control sequence is applied tothe DWN at the current time step and the problem is solved again at the next time step, following a recedinghorizon strategy [4]. There are several examples of MPC applied to water networks in the literature, see e.g.,[5, 6, 7, 8, 9] and references therein.

Among the aforementioned references, a common approach used to cope with perturbed systems is torely on the so-called certainty equivalence property [10], which in the MPC framework leads to a perturbednominal deterministic MPC strategy, also named certainty-equivalent MPC (CE-MPC). This strategy ad-dresses perturbed systems by considering nominal models that do not include the uncertainty. Hence, theexpected value of system inputs will lead to an average performing system. In the case of linear systems withuniformly distributed scenarios, the certainty equivalence property holds [11] and this strategy is optimal.

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Nevertheless, this may not be the case due to factors such as the presence of nonlinearities. Hence, the CE-MPC is usually complemented with a (de)tuning of the controller. Although, in one hand, a frequent violationof soft constraints can occur, on the other hand, infeasible solutions would result if the constraints were harddue to the ignored effects of future uncertainty. Nevertheless, there exist other MPC schemes reported in theliterature that aim to ensure robust stability and compliance with constraints in the presence of stochasticdisturbances, see e.g., [12, 13]. As summarized in [13], alternative approaches of MPC for stochastic systemsare based on min-max MPC, tube-based MPC, and stochastic MPC (SMPC). The first two approaches areoriented to ensure worst-case robustness and consequently are conservative, while the third approach relieson stochastic programming (SP) techniques to offer a probabilistic constraint fulfilment [14]. Since someviolations are allowed, the solutions obtained are less conservative and hence the performance is better interms of cost from the objective function. In this way, disturbances are modelled as random variables, andthe control problem is stated by using the expected value of the system variables, i.e., states and controlinputs.

The stochastic approach is a mature theory in the field of optimization [15], but renewed attention hasbeen given to it due to its great potential in control applications, see e.g., [16] and references therein. Fromthe wide range of SMPC methods, this paper focuses on three specific techniques, namely: chance-constrainedMPC (CC-MPC), tree-based MPC (TB-MPC), and multiple-scenario MPC (MS-MPC), also called MultipleMPC in [17].

CC-MPC is a stochastic control strategy that provides robustness in terms of probabilistic (chance)constraints, such that the probability of violation of any operational requirement or physical constraintis below a prescribed value [18]. Some works that address the CC-MPC approach in water systems are[19, 20, 21] and references therein. On the other hand, in the TB-MPC, uncertainty is addressed by consideringsimultaneously a set of possible disturbance scenarios modelled as a rooted tree, which branches along theprediction horizon with a common initial value of the disturbances, see, e.g., [22, 23, 24]. This techniquecomputes a set of control input sequences, one per each scenario in the disturbance tree, but only applies thefirst element from all sequences, which is the same for all the possible scenarios considered. At the next timeinstant, the control inputs are recalculated and only the value of the control sequence that corresponds tothe first time instant is applied. The MS-MPC follows a similar approach, but it computes a single controlsequence instead of a tree of control inputs. To do so, the MS-MPC takes into account also different possibleevolutions of the process disturbances but not necessarily in the form of a tree [12]. This approach was used in[17] for designing an MPC controller for drainage systems. In this way, both the TB-MPC and the MS-MPCincorporate robustness by considering several disturbance scenarios in a single optimization problem.

The main contribution of this paper consists in the design and assessment of the three aforementionedstochastic controllers, i.e., CC-MPC, TB-MPC, and MS-MPC, applied to the operational management of areal DWN, i.e., the Barcelona DWN. These approaches offer a compromise between economic costs, reliabilityand computational burden. A discussion of the advantages and weaknesses of these control approaches in thesense of tractability and performance is presented when applied to two case studies, i.e., a small-scale and alarge-scale model of the Barcelona DWN. This paper is an extension of the results presented in [21], whichdetails the CC-MPC controller design for DWN, and [25], which provides a comparison between TB-MPCand CC-MPC applied to DWN.

The remainder of the paper is organized as follows. Section 2 describes the case studies based on theBarcelona DWN and discusses the modelling of water demand. Section 3 describes the DWN control problemand introduces the CE-MPC formulation. Section 4 describes the formulation of the CC-MPC, TB-MPCand MS-MPC approaches. Section 5 discusses the results of the application of the three proposed SMPCapproaches to the case studies via simulation. Finally, the conclusions are drawn in Section 6.

Notation. Throughout this paper, R, Rn, Rm×n and R+ denote the field of real numbers, the set ofcolumn real vectors of length n, the set of m by n real matrices and the set of non-negative real numbers,respectively. Moreover, Sn++ denotes the set of positive definite matrices of dimension n, while Z+ denotes the

set of non-negative integer numbers including zero. Z[a,b] , {x ∈ Z+ | c1 ≤ x ≤ c2} for some c1, c2 ∈ Z+ and

Z≥c , {x ∈ Z+ | x ≥ c} for some c ∈ Z+. For a vector x ∈ Rn, x(i) denotes the i-th element of x. Similarly,X(i) denotes the i-th row of a matrix X ∈ Rn×m. Additionally, ‖·‖Z denotes the weighted 2-norm of a vector,

i.e., ‖x‖Z = (x>Zx)1/2. If not otherwise indicated, all vectors are column vectors. Transposition is denotedby superscript >, similarity is denoted by ∼, and the operators <,≤,=, >,≥ denote element-wise relations

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of vectors. Furthermore, 0 denotes a zero column vector and I the identity matrix, both of appropriatedimensions.

2 Case Studies Description and Preliminaries

The MPC approaches presented in this paper are assessed with two representative case studies based on theBarcelona DWN.

A DWN must satisfy water demands and guarantee service reliability by making optimal use of watersources and network components in order to minimize economic costs. The water network operates as a full-interconnected system driven by endogenous and exogenous flow demands. In the Barcelona DWN, water istaken from both superficial and underground sources. Flows coming from sources are regulated by pumpsor valves. After being extracted from sources, water is purified up to levels suitable for human consumptionin four water treatment plants (WTP). The water flow from any of the sources is limited and has costsassociated to the extraction and the treatment required. The DWN is divided in two management layers:the transport network, which links the water treatment plants with the reservoirs located all over the city,and the distribution network, which is sectored in sub-networks, linking reservoirs directly to consumers. Inthis work, each sector of the distribution network is considered as a pooled demand to be satisfied by thetransport network.

The two systems under study have been extracted from the Barcelona transport network. The first casestudy consists in a sector model and the second one is an aggregate model of the whole network. They differmainly in the size of the network flow problems and the number of constitutive elements:

• The sector network considers only a small-scale subsystem related to a portion of the overall DWN (seeFig. 1). This case study considers 2 water sources, 3 tanks, 6 flows controlled by valves and pumps, 4demand nodes and 2 intersection nodes.

• The aggregate network represents a simplification from the original DWN, where sets of elements areaggregated in a single element in order to reduce the size of the original model (see Fig. 2). It consists of9 water sources, 17 tanks, 61 flows controlled by valves and pumps, 25 demand nodes and 11 intersectionnodes.

2.1 Demand Modelling and Scenario Generation

In DWNs, the uncertainty is generally introduced by the stochastic behaviour of water consumers. Therefore,a proper demand modelling is required to achieve an acceptable water supply service level. For the case studiesof this paper, time series forecasting based on auto-regressive integrated moving average (ARIMA) modelsare used due to its ability to capture complex linear dynamics from historical data [26]. In this way, it ispossible to generate artificial scenarios with similar statistical properties to those obtained from historicaldata.

A correct sampling of scenarios is essential for developing the proposed SMPC approaches. For theCC-MPC approach, ARIMA models are used to generate a large number of possible demand scenarios byMonte Carlo sampling for a given time horizon N ∈ Z+; the mean demand is then used for the controllerdesign. For the MS-MPC approach, a set of Ns ∈ Z+ water demand scenarios is generated and used.Increasing the number of scenarios allows the controller to gain robustness but at the expense of additionalcomputational effort and economic performance losses. MS-MPC is generally over-conservative, because itdoes not consider the controller capacity to adapt to the new observations of the uncertainty and reformulateits controller structure at each time instant. To cope with this drawback, a representative subset of scenariosmight be chosen using scenario reduction algorithms [27, 28]. Moreover, the reduced set of scenarios can betransformed into a rooted tree of possible evolutions of the demand [29] that can be used with the TB-MPCapproach, see, e.g., [30, 31]. More specifically, a reduction of the initial number of scenarios into a rooted treeof Nr scenarios, obtained by generating an ensemble forecast tree, only reduces the number of scenarios thathave similar features with their adjacent scenarios. The disturbances tree remains the dominant scenarios.The rationale behind this approach is that uncertainty spreads with time, i.e., it is possible to predict more

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Figure 1: Sector diagram of the Barcelona DWN

accurately the demand evolution in a short time horizon than in a large one. Besides, TB-MPC takes intoaccount, within the optimization problem, the MPC capacity to adapt, i.e., a control input sequence iscalculated for each branch of the tree at each time step, by implementing the so-called Multi-stage StochasticProgramming, as pointed in [22].

3 DWN Control Problem Statement

This section introduces the CE-MPC formulation, including the system defined by time-invariant state-spacelinear model in discrete time, its goals and constraints.

3.1 Control-oriented Model

The system model may be described considering the volume of water in tanks as the state variables x ∈ Rn,the flow through the actuators as the manipulated inputs u ∈ Rm, and the demanded flows as additivemeasured disturbances d ∈ Rp. The control-oriented model of the network is described by the followingequations for all time instant k ∈ Z+:

xk+1 = Axk +Buk +Bpdk, (1a)

0 = Euuk + Eddk, (1b)

where (1a) and (1b) describe the mass balance equations for storage tanks and intersection nodes, respectively.Moreover, A ∈ Rn×n, B ∈ Rn×m, Bp ∈ Rn×p, Eu ∈ Rq×m and Ed ∈ Rq×p, are time-invariant matricesdictated by the network topology.

Assumption 1 The states in x and the demands in d are measured at any time instant k ∈ Z+.

Assumption 2 The pair (A,B) is controllable and (1b) is reachable1, provided that q ≤ m with rank(Eu) =q.

1If q < m, then multiple solutions exist, so uk should be selected by means of an optimization problem. Equation (1b)implies the possible existence of uncontrollable flows dk at the junction nodes. Therefore, a subset of the control inputs will berestricted by the domain of some flow demands.

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Figure 2: Aggregate model of the Barcelona DWN

Assumption 3 The realization of disturbances at the current time instant k may be decomposed as

dk = dk + ek, (2)

where dk ∈ Rnd is the vector of expected disturbances, and ek ∈ Rnd is the vector of forecasting errorswith non-stationary uncertainty and a known (or approximated) quasi-concave probability distribution D.Therefore, the stochastic nature of each jth row of dk is described by d(j),k ∼ Di(d(j),k,Σ(e(j),k)), where d(j),k

denotes its mean, and Σ(e(j),k) its variance.

Assumption 4 The demands are bounded, i.e., dk ∈ D, for all k ∈ Z+, and input-disturbance dominanceconstraints hold, i.e., BdD ⊆ −BU and EdD ⊆ −EuU.

The system is subject to storage and flow capacity hard constraints considered here in the form of convexpolyhedra defined as

xk ∈ X , {x ∈ Rn | Gx ≤ g}, (3a)

uk ∈ U , {u ∈ Rm | Hu ≤ h}, (3b)

for all k, where G ∈ Rrx×n, g ∈ Rrx , H ∈ Rru×m, h ∈ Rru , being rx ∈ Z+ and ru ∈ Z+ the number of stateand input constraints, respectively.

Note that in (1b) a subset of controlled flows are directly related with a subset of uncontrolled flows. Hence,u does not take values in Rm but in a linear variety. This latter observation, in addition to Assumption 2,can be exploited to develop an affine parametrization of control variables in terms of a minimum set ofdisturbances as shown in [21, Appendix A], mapping control problems to a space with a smaller decision

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vector and with less computational burden due to the elimination of the equality constraints. Thus, thesystem (1) can be rewritten as

xk+1 = Axk + Buk + Bddk, (4)

and the input constraint (3b) replaced with a time-varying restricted set defined as

Uk , {u ∈ Rm−q |HP1u ≤ h−HP2dk} ∀k ∈ Z+, (5)

where B ∈ Rn×(m−q), Bd ∈ Rn×p, P1 ∈ Rm×(m−q) and P2 ∈ Rm×p, are selection and permutation matrices(see [21, Appendix A] for details). The sets Uk are non-empty for all k due to Assumption 4.

3.2 Control Problem Statement

The goal is to design a control law that minimises a (possibly multi-objective) convex stage cost `(k, x, u) :Z+ × X × Uk → R+, which bears a functional relationship to the economics of the system operation. Letxk ∈ X be the current state and let dk = {dk+i|k}i∈Z[0,N−1]

be the sequence of disturbances over a givenprediction horizon N ∈ Z≥1. The first element of this sequence is measured, i.e., dk|k = dk, while the rest ofthe elements are estimates of future disturbances computed by an exogenous system and are available at eachtime step k ∈ Z+. Hence, the MPC controller design is based on the solution of the following finite-horizonoptimization problem:

minuk

N−1∑i=0

`(k + i, xk+i|k, uk+i|k), (6a)

subject to:

xk+i+1|k = Axk+i|k + Buk+i|k + Bddk+i|k, ∀i ∈ Z[0,N−1] (6b)

xk+i|k ∈ X, ∀i ∈ Z[1,N ] (6c)

uk+i|k ∈ Uk+i, ∀i ∈ Z[0,N−1] (6d)

xk|k = xk. (6e)

Assuming that (6) is feasible, i.e., there exists a non-empty control input sequence uk = {uk+i|k}i∈Z[0,N−1],

then the receding horizon philosophy and the model back-transformation commands to apply the controlinput

uk = κN (k, xk,dk) = P1u∗k|k + P2dk. (7)

This procedure is repeated at each time instant k, using the current measurements of states and disturbancesand the most recent forecast of these latter over the next future horizon.

4 SMPC approaches applied to DWN

Given the stochastic nature of future disturbances, the prediction model (4) involves additive uncertainty,i.e., the water demand, which is an unmanipulated input. In this way, the change in water demand dependson the uncertainty of external factors, see, e.g., [32, 33]. Therefore, the historical water demand, in animplicit manner, takes into account historical data related to demographic, meteorological, and hydrologicalconditions, among other factors. In this sense, the compliance of constraints for a given control input cannotbe ensured. Consequently, the use of SMPC strategies may allow to establish a trade-off between robustnessand performance. In this context, three SMPC approaches are formulated below.

4.1 CC-MPC

Since the optimal solution to (6) does not always imply feasibility of the real system, it is appropriate to relaxthe original constraints in (21c) with probabilistic statements in the form of the so-called chance constraints.In this way, state constraints are required to be satisfied with a predefined probability to manage the reliabilityof the system. Considering the form of the state constraint set X, there are two types of chance constraintsaccording to the definitions below.

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Definition 1 (Joint chance constraint) A (linear) state joint chance constraint is of the form

P[G(j)x ≤ g(j) , ∀j ∈ Z[1,rx]] ≥ 1− δx, (8)

where P denotes the probability operator, δx ∈ (0, 1) is the risk acceptability level of constraint violation forthe states, and G(j) and g(j) denote the jth row of G and g, respectively. This requires that all rows j haveto be jointly fulfilled with the probability 1− δx.

Definition 2 (Individual chance constraint) A (linear) state individual chance constraint is of the form

P[G(j)x ≤ g(j)] ≥ 1− δx,j , ∀j ∈ Z[1,rx], (9)

which requires that each jth row of the inequality has to be fulfilled individually with the respective probability1− δx,j, where δx,j ∈ (0, 1).

Both forms of constraints are useful to measure risks, hence, their selection depends on the application. Allchance-constrained models require prior knowledge of the acceptable risk δx associated with the constraints.A lower risk acceptability implies a harder constraint. This article is concerned with the use of joint chanceconstraints since they can express better the management of the overall reliability in a DWN. In general, jointchance constraints lack from analytic expressions due to the involved multivariate probability distribution.Nevertheless, sampling-based methods, numeric integration, and convex analytic approximations exists, seee.g., [15] and references therein. Here, (8) is approximated following the results in [34, 35] by upper boundingthe joint constraint and assuming a uniform distribution of the joint risk among a set of individual chanceconstraints that are later transformed into equivalent deterministic constraints under Assumption 5.

Assumption 5 Each demand in d ∈ Rp follows a log-concave univariate distribution, whose stochasticdescription is known.

Given the dynamic model in (4), the stochastic nature of the demand vector d makes the state vectorx ∈ Rn to be also a stochastic variable. Then, let the cumulative distribution function of the constraint bedenoted as

FGx(g) , P[{G(1)x ≤ g(1), . . . , G(rx)x ≤ g(rx)

}]. (10)

Defining the events Cj ,{G(j)x ≤ g(j)

}for all j ∈ Zrx1 , and denoting their complements as Ccj ,

{G(j)x > g(j)

},

then it follows that

FGx(g) = P [C1 ∩ . . . ∩ Crx ] (11a)

= P[(Cc1 ∪ . . . ∪ Ccrx)c

](11b)

= 1− P[(Cc1 ∪ . . . ∪ Ccrx)

]≥ 1− δx. (11c)

Taking advantage of the union bound, the Boole’s inequality allows to bound the probability of the secondterm in the left-hand side of (11c), stating that for a countable set of events, the probability that at leastone event happens is not higher than the sum of the individual probabilities [34]. This yields

P

rx⋃j=1

Ccj

≤ rx∑j=1

P[Ccj]. (12)

Applying (12) to the inequality in (11c), it follows that

rx∑j=1

P[Ccj]≤ δx ⇔

rx∑j=1

(1− P [Cj ]) ≤ δx. (13)

At this point, a set of constraints arises from previous result as sufficient conditions to enforce the jointchance constraint (8), by allocating the joint risk δx in separate individual risks denoted by δx,j , j ∈ Zrx1 .

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These constraints are

P [Cj ] ≥ 1− δx,j , ∀j ∈ Zrx1 , (14)rx∑j=1

δx,j ≤ δx, (15)

0 ≤ δx,j ≤ 1, (16)

where (14) forms the set of rx resultant individual chance constraints, which bounds the probability thateach inequality of the receding horizon problem may fail; and (15) and (16) are conditions imposed to boundthe new single risks in such a way that the joint risk bound is not violated. Any solution that satisfies theabove constraints, is guaranteed to satisfy (8). As proposed in [35], assigning, e.g., a fixed and equal valueof risk to each individual constraint, i.e., δx,j = δx/rx for all j ∈ Z[1,rx], then (15) and (16) are satisfied.

Remark 1 The single risks δx,j, j ∈ Z[1,rx], might be considered as new decision variables to be optimised,see e.g., [36]. This should improve the performance but at the cost of more computational burden due to thegreater complexity and dimensionality of the optimization task. Therefore, as DWNs are often large-scalesystems, the uniform risk allocation policy is adopted to avoid overloading of the optimization problem. ♦

After decomposing the joint constraints into a set of individual constraints, the deterministic equivalentof each separate constraint may be used given that the probabilistic statements are not suitable for algebraicsolution. Such deterministic equivalents might be obtained following the results in [37]. Assuming a known(or approximated) quasi-concave probabilistic distribution function for the effect of the stochastic disturbancein the dynamic model (4), it follows that

P[G(j)xk+1 ≤ g(j)

]≥ 1− δx,j ⇔ FG(j)Bddk

(g(j) −G(j)(Axk + Buk)) ≥ 1− δx,j

⇔ G(j)(Axk + Buk) ≤ g(j) − F−1

G(j)Bddk(1− δx,j), (17)

for all j ∈ Z[1,rx], where FG(j)Bddk(·) and F−1

G(j)Bddk(·) are the cumulative distribution and the left-quantile

function of G(j)Bddk, respectively. Hence, the original state constraint set X is contracted by the effect ofthe rx deterministic equivalents in (17) and replaced by the stochastic feasibility set given by

Xs,k , {xk ∈ Rn |∃uk ∈ Uk, such that

G(j)(Axk + Buk) ≤ g(j) − F−1

G(j)Bddk(1− δx,j), ∀j ∈ Z[1,rx]},

for all k ∈ Z+. From convexity of G(j)xk+1 ≤ g(j) and Assumption 5, it follows that the set Xs,k is convexwhen non-empty for all δx,j ∈ (0, 1) [38] and most quasi-concave distribution function. For some particulardistributions, e.g., Gaussian, convexity is retained for δx,j ∈ (0, 0.5]

In this way, the reformulated predictive controller solves the following deterministic equivalent optimiza-tion problem for the expectation E[·] of the cost function in (6a):

minuk

N−1∑i=0

E[`(k + i, xk+i|k, uk+i|k)], (18a)

subject to:

xk+i+1|k = Axk+i|k + Buk+i|k + Bddk+i|k, ∀i ∈ Z[0,N−1] (18b)

G(j)(Axk+i|k + Buk+i|k) ≤ g(j) − zk,j(δx), ∀i ∈ Z[0,N−1], ∀j ∈ Z[1,rx], (18c)

uk+i|k ∈ Uk+i, ∀i ∈ Z[0,N−1] (18d)

xk|k = xk, (18e)

where uk = {uk+i|k}i∈Z[0,N−1]is the sequence of controlled flows, dk|k = dk is the current demand and dk+i|k

are expected future demands computed at time instant k ∈ Z+ for i-steps ahead with i ∈ Z[1,N−1]. Moreover,

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Figure 3: Reduction of a disturbance fan (left) of equally probable scenarios into a rooted scenario-tree(right).

zk,j(δx) , F−1

G(j)Bddk+i

(1− δx

nc

), where nc ∈ Z≥1 is the number of total individual state constraints along the

prediction horizon, i.e., nc = rxN . Since nc depends not only on the number of state constraints rx but alsoon the value of N , the decomposition of the original joint chance constraint within the MPC algorithm couldlead to a large number of constraints. This fact reinforces the use of a fixed risk distribution policy for DWNcontrol problems, in order to avoid the addition of a large number of new decision variables to be optimized.

Remark 2 It turns out that most (not all) probability distribution functions used in different applications,e.g., uniform, Gaussian, logistic, Chi-squared, Gamma, Beta, log-normal, Weibull, Dirichlet, Wishart, amongothers, share the property of being log-concave. Then, their corresponding quantile function can be computedoff-line for a given risk acceptability level and used within the MPC convex optimization. ♦

4.2 TB-MPC

CC-MPC might be conservative if the probabilistic distributions of the stochastic variables are not properlycharacterized or do not have a log-concave form. Therefore, this section presents the TB-MPC strategythat relies on scenario trees to approximate the original problem, dropping Assumption 5. The approachfollowed by TB-MPC is based on modelling the possible scenarios of the disturbances as a rooted tree, asseen in Fig. 3. This means that all the scenarios start from the same measured disturbance value. Fromthat initial point, the scenarios must remain equal until the point in which they diverge behaving differently.This point is called a bifurcation point, as described in [27]. Each node of the tree has a unique parent andcan have many children. The total number of children at the last stage corresponds to the total number ofscenarios. The probability of a scenario is the product of probabilities of each node in that scenario. Recallthat a large number of scenarios might improve the robustness of the TB-MPC approach but at the costof additional computational burden and economic performance losses. Hence, a trade-off must be achievedbetween performance and computational burden.

Notice that before a bifurcation point, the evolution followed by the disturbance cannot be anticipatedbecause different evolutions are possible. For this reason, the controller has to calculate control inputs thatare valid for all the scenarios in the branch. Once the bifurcation point has been reached, the uncertainty issolved and the controller can calculate specific control inputs for the scenarios in each of the new branches[39]. Hence, the outcome of TB-MPC is not a single sequence of control inputs, it is a tree with the samestructure of that of the disturbances. Only the first element of this tree is applied (the root) and the problemis repeated in a receding horizon fashion.

The easiest way to understand the optimization problem associated to TB-MPC is to solve as manyinstances of Problem (6) as the number Nr ∈ Z+ of scenarios considered, but formally it is a multi-stagestochastic programming problem that should be and solved as a big optimization problem for all the scenarios.Due to the increasing uncertainty, it is necessary to include non-anticipativity constraints [40] into the MPC

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formulation so that the calculated input sequence is always ready to face any possible future bifurcation inthe tree [41]. TB-MPC constrains the control inputs relative to a scenario and its parent from the root attime instant k to the branching point. In this sense, it is necessary to define the tree structure, at each timestep k over the horizon N , by means of two functions for each scenario: a parent function P (·), defined fromZ[1,Nr] to Z[1,Nr], which indicates from which scenario it branches out; and a branching point function BP (·),defined from Z[1,Nr] to Z[k,k+N ], which indicates the time when the scenario emerges as different from itsparent. For more details, see [27] and references therein.

More specifically, if dak = {dak|k, dak+1|k, . . . , d

ak+N |k} and dbk = {dbk|k, d

bk+1|k, . . . , d

bk+N |k} are two distur-

bance sequences corresponding respectively to the forecast scenarios a, b ∈ Z[1,Nr], then the non-anticipative

constraint uak+i|k = ubk+i|k must be satisfied whenever the parent of the scenario a was b, that is P (a) = b,

and the scenario a has not emerged yet at time instant k + i, i.e., k + i < BP (a) for any i ∈ Z[0,N ] in order

to guarantee that for all j ∈ Z[1,Nr] the control input sequences uj = {ujk+i|k}i∈Z[0,N−1]do not depend on the

unobserved stochastic variables, as established in [22].In this way, the TB-MPC controller should solve the following optimization problem at each time instant

k ∈ Z+, accounting for the Nr demand scenarios, each with probability pj ∈ (0, 1] satisfying∑Nr

j=1 pj = 1:

minuj

k

Nr∑j=1

pj

(N−1∑i=0

`(k + i, xjk+i|k, ujk+i|k)

), (19a)

subject to:

xjk+i+1|k = Axjk+i|k + Bujk+i|k + Bddjk+i|k, (19b)

xjk+i+1|k ∈ X, (19c)

ujk+i|k ∈ Ujk+i, (19d)

xjk|k = xk, djk|k = dk, (19e)

uak+i|k = ubk+i|k when

{P (a) = b,k + i < BP (a)

∀a, b ∈ Z[1,Nr], (19f)

for all i ∈ Z[0,N−1] and all j ∈ Z[1,Nr], where Ujk+i , {uj ∈ Rm−q |HPM1uj ≤ h−HPM2d

jk+i}.

Remark 3 In practice, Problem (19) can be solved more efficiently by using a transformation matrix thatremoves the redundancy of the optimization variables due to non-anticipative constraints (19f). In thismanner, the number of optimization variables and constraints decrease [42]. ♦

The control signal is the same for all scenarios at the current time step since these scenarios start withthe measured disturbance and equality constraints are formulated. The optimization problem meets theseconstraints and computes the control input under economic policies. If the solver is not able to calculate thecontrol inputs by infeasibility, these equality constraints can become soft-constraints and penalize deviations.In addition, note that uncertainty spreads with time, i.e., the estimation of the disturbances at the first timeinstant can be carried out with a certain degree of accuracy. Scenarios tend to diverge with time because theuncertainty grows as well.

Remark 4 Another way to solve the optimization problem in (19) is to treat each scenario independently byconsidering distributed optimization and attain a coordinated solution using a distributed control scheme, asshown in [41]. ♦

4.3 MS-MPC

The optimization based on scenarios provides an intuitive way to approximate the solution of the stochasticoptimization problem. The MS-MPC approach can be implemented similarly to the TB-MPC. The differencebetween both approaches is that while TB-MPC calculates as many control sequences as scenarios considered,MS-MPC optimizes a single control sequence valid for all the scenarios.

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The MS-MPC approach is computationally efficient since its solution is based on a deterministic convexoptimization, even when the original problem is not [43]. Thus, a stochastic optimization is solved in adeterministic way. One of its advantages is that it is possible to calculate the risk acceptability level (δx) ofconstraint violation as a function of the number of scenarios considered for any violation level υ ∈ [0, 1] [28].In particular, the probability of the predicted states to satisfy the constraints is given by

P[xi+1 ∈ X , ∀i ∈ ZN−10 ] ≥ 1− δx, (20a)

where

δx =

∫ 1

0

[min{1,(R+ ρ1 − 1

R

)R+ρ1−1∑j=0

(K

j

)υj(1− υ)K−j}dυ]. (20b)

As it can be seen, the risk acceptability level δx is a function of K ∈ Z+ that establishes the total number ofscenarios generated that represent the evolutions of the water demand, R ∈ Z+ that represents the number ofdiscarded scenarios (R < K), and ρ1 which is a parameter related to the algebraic properties of the problem[43]. Notice that Ns = K −R, that is, the number of scenarios considered in the FHOP.

Remark 5 Note that the calculation of probabilistic bounds for TB-MPC is beyond the scope of this work.However, given that the performance of TB-MPC is better than that of MS-MPC (the controller is executedin closed loop from a control viewpoint in TB-MPC), the same probabilistic bounds could be used for both,MS-MPC and TB-MPC, at least as a practical rule. ♦

The MS-MPC controller should solve the following FHOP, accounting for the Ns ∈ Z+ scenarios:

minuj

k

Nr∑j=1

pj

(N−1∑i=0

`(k + i, xjk+i|k, uk+i|k)

), (21a)

subject to:

xjk+i+1|k = Axjk+i|k + Buk+i|k + Bddjk+i|k, (21b)

xjk+i+1|k ∈ X, (21c)

uk+i|k ∈ Uk+i, (21d)

xjk|k = xk, djk|k = dk, (21e)

for all i ∈ Z[0,N−1] and all j ∈ Z[1,Ns], where pj is again the probability of occurrence of each consideredscenario [17].

It should be noticed that this approach reduces the number of decision variables in the FHOP comparedto the TB-MPC approach, but it might increase the computational time since having less degrees of freedomincreases the complexity of complying simultaneously with all the considered scenarios.

Remark 6 One way to address the complexity of the MS-MPC optimization problem is to follow the ideaspresented in [17, 44], where only three disturbance scenarios are considered, i.e., the best, the worst and theaverage case, each one with its probability of occurrence. This is a practical way to relax the computationalburden. ♦

In this paper, the simplification highlighted in Remark 6 is adopted. In this way, the MS-MPC approachconsiders the solution of (21) with Ns = 3.

5 Results

The formulation of the SMPC problems for the case studies considered in this paper addresses the design of acontrol law that (i) minimizes the economic operational cost, (ii) guarantees the availability of enough waterto satisfy the demand and (iii) operates the network with smooth variations of the flow through actuators.These objectives can be expressed quantitatively by the following performance indicators2 for all time steps

2The performance indicators considered in this paper may vary or be generalized with the corresponding manipulation toinclude other control objectives.

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k ∈ Z+:

`E(xk, uk; cu,k) , c>u,kWe uk∆t, (22a)

`S(xk; sk) ,

{(xk − sk)>Ws(xk − sk) if xk ≤ sk0 otherwise,

(22b)

`∆(∆uk) , ∆u>kW∆u ∆uk. (22c)

The first objective, `E(xk, uk; cu,k) ∈ R≥0, represents the economic cost of network operation at time stepk, which depends on a time-of-use pricing scheme driven by a time-varying price of the flow through theactuators cu,k , (c1 + c2,k) ∈ Rm−q+ , which in this application takes into account a fixed water productioncost c1 ∈ Rm+ and a water pumping cost c2,k ∈ Rm+ that changes according to the electricity tariff (assumedperiodically time-varying). All prices are given in economic units per cubic meter (e.u./m3). The secondobjective, `S(xk; sk) ∈ R≥0 for all k, is a performance index that penalizes the amount of water volumegoing below a given safety threshold sk ∈ Rn in m3, which is desired to be stored in tanks and satisfiesthe condition xmin ≤ sk ≤ xmax. Note that this safety objective is a piecewise continuous function, butit can be redefined as `S(ξk;xk, sk) , ξ>k Ws ξk, accompanied with two additional convex constraints, i.e.,xk ≥ sk − ξk and ξk ∈ Rn+, for all k, being ξk a slack variable. The minimal volume of water requiredin a tank is given by its net demand, hence, sk = −Bpdk for all k. The last objective, `∆(∆uk) ∈ R≥0,

represents the penalization of control signal variations ∆uk , uk − uk−1 ∈ Rm−q. The inclusion of thislatter objective aims to extend actuators life and assure a smooth operation of the dynamic network flows.Furthermore, We ∈ Sm−q++ , Ws ∈ Sn++ and W∆u ∈ Sm−q++ are matrices that weight each decision variable intheir corresponding cost function.

To achieve the control task, the above predefined objectives are aggregated in a multi-objective stage costfunction, which depends explicitly on time due to the time-varying parameters of the involved individualobjectives. The overall stage cost is defined for all k ∈ Z+ as

`(k, xk, uk, ξk) , λ1`E(xk, uk; cu,k) + λ2`∆(∆uk) + λ3`S(ξk;xk, sk), (23)

where λ1, λ2, λ3 ∈ R+ are scalar weights that allow to prioritize the impact of each objective involved in theoverall performance of the network. These weights are assumed to be fixed by the managers of the DWN.

Numerical results of applying the three different SMPC approaches (CC-MPC, TB-MPC and MS-MPC)to the Barcelona DWN case studies are summarized in Tables 1, 2 and 3.

Simulations have been carried out over a time horizon of eight days, i.e., ns = 192 hours, with a samplingtime of one hour. The patterns of the water demand in this paper were synthesized from real values measuredin the considered demand of the Barcelona DWN between July 23th and July 27th, 2007. Initial conditions,i.e., source capacities, initial volume of water at tanks and starting demands, are set a priori accordingto real data. The weights of the cost function (23) are λ1 = 100, λ2 = 10, and λ3 = 1; these valuesallow to prioritize the impact of each objective involved in the overall performance of the network. Theprediction horizon is selected as N = 24 hours, due to the periodicity of disturbances. The formulation ofthe optimization problems and the closed-loop simulations have been carried out using MATLAB R2012a(64 bits) and CPLEX solver, running in a PC Core i7 at 3.2 GHz with 16 GB of RAM.

The key performance indicators used to assess the aforementioned controllers are defined as follows:

Φ1 ,24

ns

ns∑k=1

`k, (24a)

Φ2 , |{k ∈ Z[1,ns] | xk < −Bpdk

}|, (24b)

Φ3 ,ns∑k=1

nx∑i=1

max{0,−Bp(i)dk − xk(i)}, (24c)

Φ4 ,1

ns

ns∑k=1

tk, (24d)

where Φ1 is the average daily multi-objective cost with `k given by (23), Φ2 is the number of time instantswhere water demands are not satisfied, Φ3 is the accumulated volume of water demand that was not satisfied

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Table 1: Comparison of the CC-MPC, TB-MPC and MS-MPC applied to the sector model of the DWN.

CC-MPC TB-MPC MS-MPC

δ Φ1 Φ2 Φ3 Φ4 Φ1 Φ2 Φ3 Φ4 Nr Φ1 Φ2 Φ3 Φ4 Ns δ

0.3 58535.80 0 0 0.091958397.14 0 0 1.2548 5

60831.33 0 0 16.1547 107 0.358515.40 1 0.4813 1.9701 1058589.15 1 4.155 3.0145 14

0.2 58541.19 0 0 0.70758515.37 0 0 3.2218 19

69342.48 0 0 17.2401 129 0.258678.12 2 0.7443 7.5362 3858647.27 4 4.0329 27.0914 75

0.1 58558.29 0 0 0.71658705.28 1 0.2136 35.6572 107

66011.29 0 0 21.7521 149 0.158713.15 0 0 42.7024 12958761.98 0 0 54.4587 149

over the simulation horizon ns, and Φ4 is the average time in seconds required to solve the MPC problem ateach time instant k ∈ Z[1,ns].

The effect of considering different levels of joint risk acceptability was analysed for the CC-MPC approachusing δ = {0.3, 0.2, 0.1}. In the same way, the size of the set of scenarios selected for the MS-MPC isestablished by using (20) to guarantee the same risk levels of the CC-MPC approach. In this way, the totalnumber of scenarios that represents the evolution of the water demand in the considered simulation timefor the MS-MPC was K = 192. Likewise, the TB-MPC approach was applied considering different sizes forthe set of reduced scenarios, i.e., Nr = {5, 10, 19, 38, 75, 107, 129, 149}. The last three Nr scenarios allow tocompare the behaviour between MS-MPC and TB-MPC, while the remaining scenarios were used to analysethe performance with respect to a small number of scenarios.

Table 1 summarizes the results of applying the SMPC approaches to the sector model of the BarcelonaDWN presented in Fig. 1. The different values of δ in the CC-MPC approach highlight that both reliabilityand control performance are conflicting objectives, i.e., the inclusion of safety mechanisms in the controllerincreases the reliability of the DWN in terms of demand satisfaction, but also the cost of its operation. Themain advantage of the CC-MPC is its formal methodology, which leads to obtain optimal safety constraintsthat tackle uncertainties and allow to achieve a specified global service level in the DWN. Moreover, theCC-MPC robustness is achieved with a low computational burden given that the only extra load is thecomputation of the stochastic characteristics of disturbances propagated along the prediction horizon N . Inthis way, the CC-MPC approach is suitable for real-time control of large-scale DWNs.

Regarding the TB-MPC and MS-MPC approaches, numerical results show that considering a largernumber of scenarios increments (in average) the stage cost while reducing the volume of unsatisfied waterdemand. This might be influenced by the quality of the information that remains after the reductionalgorithms and, consequently, it affects the robustness of the approach being subject of further research.

The main drawback of the TB-MPC approach is the solution average time and the computational burden.The implementation for all cases taking scenarios greater than Nr = 149 was not possible due to memoryissues. Hence, some simplification assumptions as those used in [24] or parallel computing techniques mightbe useful. Another way to address the problem generated by the computer effort is to use a MS-MPCbased on the three-scenarios approach, described in Subsection 4.3. At this point, the best, the worst and theaverage disturbance scenarios were obtained by generating 100 different possible evolutions of the disturbance,then they were lumped and averaged the 10 lowest, 10 highest, and 80 middle, respectively. It means thatthe occurrence probabilities were established as 0.1, 0.1, and 0.8 for the best, the worst, and the averagedisturbance scenario, respectively, as proposed in [17].

Additionally, Table 2 summarizes the simulation results of applying the studied SMPC approaches againto the sector model DWN. The configuration of the controllers in this case is as follows: the CC-MPC witha probability of risk of 5%, the TB-MPC reducing to Nr = 3 branched disturbances, the MS-MPC based onthe three scenarios approach (i.e., best, worst and average), and the CE-MPC with an average disturbance.On the one hand, the CE-MPC approach presents the lower cost but on the other hand it has problemswith the demand satisfaction. The TB-MPC approach presents a lower accumulated volume of unsatisfied

13

water demand compared with respect to the CE-MPC approach. The MS-MPC and CC-MPC approachesare able to satisfy the water demand required by the consumers. The CC-MPC approach presents a betterperformance regarding cost and computational time compared to the MS-MPC approach.

Table 2: Comparison of the MS-MPC, TB-MPC, CC-MPC and CE-MPC applied to the sector model DWN.

Controller Φ1 Φ2 Φ3 Φ4

CC-MPC 585401.16 0 0 0.1069TB-MPC 58425.59 1 1.2229 1.4235MS-MPC 60567.62 1 0.6965 0.5314CE-MPC 58327.55 1 0.7411 0.1041

As for the second case study, Table 3 presents the results obtained after the application of the SMPCapproaches to the aggregate model DWN, as a way to show the strengths and weaknesses of each of theaforementioned approaches applied to a larger system. For this reason, TB-MPC and MS-MPC with alarge number of scenarios, could not be applied due to memory issues. TB-MPC was implemented witha reduction to Nr = 3 branched scenarios. MS-MPC has been designed considering the three scenarios(minimum, average and maximum) as explained in the previous case study. CC-MPC is applied to thissystem with a risk probability of 5%. As it can be seen from the results, the TB-MPC approach does notoffer benefits in terms of satisfaction of water demand and computational time with this limited amount ofscenarios for the DWN aggregate model. MS-MPC presents encouraging results regarding demand satisfactionand computational time well below that TB-MPC. MS-MPC approach presents a higher average daily multi-objective cost and a computational time required to solve the FHOP around three times more regardingCC-MPC. Furthermore, MS-MPC and CC-MPC, have a good performance with respect to water demandsatisfaction. Based on the obtained results, the CC-MPC approach offers better performance in terms ofdemand satisfaction, computational time and, it presents the best behaviour with respect to the averagedaily multi-objective cost (φ1) compared with the same indicator obtained with TB-MPC and MS-MPCapproaches.

Table 3: Comparison of the MS-MPC, TB-MPC, CC-MPC and CE-MPC applied to the DWN case studyfor the aggregate DWN model.

Controller Φ1 Φ2 Φ3 Φ4

CC-MPC 1.4064 · 104 0 0 0.9056TB-MPC 1.4497 · 104 20 18.81 8.48MS-MPC 1.5267 · 104 0 0 3.24CE-MPC 1.2038 · 104 23 5.211 0.8442

As expected, the SMPC approaches have a better performance indicators with respect to CE-MPCapproach, which does not take into account the stochastic nature of water demand in the formulation ofFHOP.

6 Conclusions

In this paper, three stochastic control approaches have been assessed to deal with the optimal operationalmanagement of a DWN. The proposed stochastic control strategies are based on MPC which allows theoptimization of the objective function to ensure a reliable water supply in the most economic and safe wayby considering the uncertainty in the water consumption. Simulation results show that all the consideredmethods require less than 1 minute to solve the optimization problem in each case, much shorter than thesampling time of 1 hour. Hence, it is possible to choose an approach that shows the best performance interms of demand satisfaction, which is given by the number of time instants where water demands were notsatisfied and by the cumulated volume of non-satisfied water demand. In this sense, the results in this papershow that CC-MPC is more appropriate when requiring a low probability of constraint violation, because

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the use of TB-MPC and MS-MPC implies the inclusion of a higher number of scenarios, which hinders theapplication of these control strategies to large DWNs. However, the use of these scenarios-based approachesmay be very demanding in terms of computational time.

From a practical viewpoint, both TB-MPC and CC-MPC outperform MS-MPC. TB-MPC excels dueto its close-loop control capability and requires to build a tree to reproduce the main dynamics of thedisturbance [27, 45, 46]. CC-MPC improves the performance by allowing a probability of constraint violationand requires considering a well-known behaviour of water user demand in terms of probability distribution.Finally, MS-MPC is an alternative to cope with the uncertainty, which considers the possible evolutions ofthe water demand modelled as scenarios, by using an open-loop forecasting of disturbances. A drawback thatpresents this approach is its reduced capacity to be adapted to changes that the water demand could arise.This may result in more conservative control inputs that increase the economic costs.

Each of the approaches described have their advantages and weakness. CC-MPC presents the bestcomputational time and economic costs against the need to take account of a probability distribution functionthat represents the behaviour of the disturbances. TB-MPC and MS-MPC approaches are ways to solvethe stochastic optimization in a deterministic manner, but require a greater computational effort. FinallyTB-MPC and MS-MPC approaches can be implemented using a reduced number of scenarios at the cost oflower robustness.

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